2015/08/25 by Dimiter Prodanov, Prodanov, Dimiter
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA
paper · pdf · doi:10.48550/arxiv.1508.06086
21 pages, 1 figure
arxiv created 2015/08/25 · arxiv updated 2015/08/26
Holderian functions have strong non-linearities, which result in singularities in the derivatives. This manuscript presents several fractional-order Taylor expansions of Hölderian functions around points of non- differentiability. These expansions are derived using the concept of a fractional velocity, which can be used to describe the singular behavior of derivatives and non-differentiable functions. Fractional velocity is defined as the limit of the difference quotient of the increment of a function and the difference of its argument raised to a fractional power. Fractional velocity can be used to regularize ordinary derivatives. To this end, it is possible to define regularized Taylor series and compound differential rules. In particular a compound differential rule for Holder 1/2 functions is demonstrated. The expansion is presented using the auxiliary concept of fractional co-variation of functions.